An abstract in French for the scientific journal Comptes Rendus. Mrs Turing helped with the French and the typing.
这是用法语写的，是图灵夫人协助他翻译成法语并打印出来。
The lambda-calculus represented an elegant and powerful symbolism for mathematical processes of abstraction and generalisation.
λ算子能够非常简洁有力地对数学过程进行抽象和泛化。
The 'complex' number calculus exemplified the progress of mathematical abstraction.
复数是数学抽象化的又一个进展。
Originally, complex numbers had been introduced to combine 'real' numbers with the 'imaginary' square root of minus one, and mathematicians had agonised over the question of whether such things really 'existed'.
最初人们引入复数，将实数与虚数（比如-1的平方根）结合起来的时候，数学家们感到非常纠结，不知道这样的东西是否真的存在。
From the modern point of view, however, complex numbers were simply defined abstractly as pairs of numbers, and pictured as points in a plane.
从现代的观点来看，可以简单地把一个复数看成一个数对，它可以形象地画在平面坐标系上，
A simple rule for the definition of the 'multiplication' of two such pairs was then sufficient to generate an enormous theory.
两个数对之间有一套简单的乘法规则，这样就可以产生很强大的理论。
Riemann's work in the nineteenth century had played a large part in its 'pure' development; but it was also found to be of great usefulness in the development of physical theory.
在19世纪以来，黎曼的工作主要是在纯数学领域发挥作用，但是人们后来发现，它们在物理领域也有很多用处。
Fourier analysis, treating the theory of vibrations, was an example of this.
傅利叶分析就是一个例子。
The quantum theory developed since the 1920s went even further in according complex numbers a place in fundamental physical concepts.
20年代以来的量子理论，更加深入地应用了复数的概念。
None of these mathematical ideas are essential to what follows, although such connections between 'pure' and 'applied' were certainly relevant to a number of aspects of Alan Turing's later work.
这些数学概念，对接下来的故事来说并不重要。不过，这种纯数学和现实应用之间的关联，倒是和艾伦·图灵后来的工作很有关系。